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Invariant curves and topological entropy

We have the following qualitative result about the topological entropy of the maps tex2html_wrap_inline289 .

propo185

Proof.
For p=1 or tex2html_wrap_inline327 , the curve tex2html_wrap_inline465 is a square and tex2html_wrap_inline289 is integrable with integral I(s,y)=y+y', where tex2html_wrap_inline833 . The topological entropy is then zero. For p=2, the curve tex2html_wrap_inline465 is a circle and is integrable having the integral I(s,y)=y. On each invariant curve I=const, the map tex2html_wrap_inline843 is a rotation. The topological entropy is also zero.
For p;SPMgt;2, there exist points on the curve where the curvature is zero. A theorem of Mather [15] assures that there exists then no homotopiclly nontrivial invariant curve. Angenent's result [1] implies then that the topological entropy is non-vanishing. For p;SPMlt;2, there exist points on the table where the curvature is unbounded. A theorem of Hubacher [10] assures that there exist no invariant curves near the boundary of X. Again, Angenent's result leads to positive topological entropy.
End of the proof.


Oliver Knill, Jul 10 1998